文档介绍：Spline Wavelets 1
Theory and Algorithms for Non-Uniform Spline Wavelets
T. Lyche1),ørken2),)
Abstract. We investigate mutually orthogonal spline wavelet spaces on non-uniform
partitions of a bounded interval, addressing the existence, uniqueness and construction
of bases of minimally supported spline wavelets. The relevant algorithms for po-
sition and reconstruction are considered as well as some stability-related questions. In
addition, we briefly review the bivariate case for tensor products and arbitrary triangu-
lations. We conclude the paper with a discussion of some special cases.
§1. Introduction
Splines have e the standard mathematical tool for representing smooth shapes in
computer graphics and geometric modelling. Wavelets have been introduced more re-
cently, but are by now well established both in mathematics and in applied sciences like
signal processing and numerical analysis. The two concepts are closely related as splines
provide some of the most important examples of wavelets. Although there is an extensive
literature on cardinal spline wavelets (spline wavelets with uniform knot spacing), see [7],
relatively little has been published about spline wavelets on arbitrary, nonuniform knots,
which form the subject of this paper. These kinds of wavelets, however, are needed to
perform operations like position, reconstruction and thresholding on splines given
on a nonuniform knot vector, which typically occur in practical applications.
The flexibility of splines in modelling is due to good approximation properties, useful
geometric interpretations of the B-spline coefficients, and simple algorithms for adding and
removing knots. Full advantage of these capabilities can only be taken on general nonuni-
form knot vectors, where also multiple knots are allowed. In fact, the spline algorithms
for general knots are hardly plicated than the special ones for uniform knots.
We will see in this paper that the same is tru